reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;
reserve f for Function of A,B;

theorem
  for X being finite set st n <= card X holds ex A being finite Subset
  of X st card A = n
proof
  let X be finite set such that
A1: n <= card X;
  consider p being FinSequence such that
A2: rng p = X and
A3: p is one-to-one by Th58;
  reconsider q = p|Seg n as FinSequence by FINSEQ_1:15;
  n <= len p by A1,A2,A3,Th62;
  then
A4: len q = n by FINSEQ_1:17;
  reconsider A = rng q as Subset of X by A2,RELAT_1:70;
  q is one-to-one by A3,FUNCT_1:52;
  then card A = n by A4,Th62;
  hence thesis;
end;
